Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
The extended euclidean algorithm, a method used to find the greatest common divisor (gcd) of two numbers and determine their linear combination. Two examples are provided, one with m = 65 and n = 40, and the other with m = 1239 and n = 735.
Typology: Study notes
1 / 2
This page cannot be seen from the preview
Don't miss anything!
Example 1: m = 65, n = 40
Step 1: The (usual) Euclidean algorithm:
(1) 65 = 1 · 40 + 25 (2) 40 = 1 · 25 + 15 (3) 25 = 1 · 15 + 10 (4) 15 = 1 · 10 + 5 10 = 2 · 5
Therefore: gcd(65, 40) = 5.
Step 2: Using the method of back-substitution:
5 (4) = 15 − 10 (3) = 15 − (25 − 15) = 2 · 15 − 25 (2) = 2(40 − 25) − 25 = 2 · 40 − 3 · 25 (1) = 2 · 40 − 3(65 − 40) = 5 · 40 − 3 · 65
Conclusion: 65(︸ −︷︷3) ︸ x
y
Example 2: m = 1239, n = 735
Step 1: The (usual) Euclidean algorithm:
(1) 1239 = 1 · 735 + 504 (2) 735 = 1 · 504 + 231 (3) 504 = 2 · 231 + 42 (4) 231 = 5 · 42 + 21 42 = 2 · 21
Therefore: gcd(1239, 735) = 21.
Step 2: Using the method of back-substitution:
21 (4) = 231 − 5 · 42 (3) = 231 − 5(504 − 2 · 231) = 11 · 231 − 5 · 504 (2) = 11(735 − 504) − 5 · 504 = 11 · 735 − 16 · 504 (1) = 11 · 735 − 16(1239 − 735) = 27 · 735 − 16 · 1239
Conclusion: 1239(︸ −︷︷16) ︸ x
y
